The dynamics of a bouncing ball with a sinusoidally vibrating table revisited

Luo, Albert C. J.; Han, Ray P. S.

doi:10.1007/bf00114795

Cited by 102 publications

(47 citation statements)

References 16 publications

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“…Dynamics of the ball from an impact to the next impact can be described by the following Poincaré map in nondimensional form [13] (see also Ref. [14] where analogous map was derived earlier and Ref. [15] for generalizations of the bouncing ball model):…”

Section: Bouncing Ball: a Simple Motion Of The Tablementioning

confidence: 99%

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Simple Model of Bouncing Ball Dynamics

Okniński

Radziszewski

2012

Differ Equ Dyn Syst

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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, 2 -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.

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Section: Bouncing Ball: a Simple Motion Of The Tablementioning

confidence: 99%

“…The table's motion has been usually assumed in form Y s (T ) = sin(T ), cf. [14,15] and references therein. In this case it is basically impossible to solve the Eq.…”

Section: Bouncing Ball: a Simple Motion Of The Tablementioning

confidence: 99%

Simple Model of Bouncing Ball Dynamics

Okniński

Radziszewski

2012

Differ Equ Dyn Syst

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“…For the nondissipative version, the system basically behaves like the standard map in a local approximation [2,9], where some of the previous findings concerning the ballistic transport and accelerator modes (AMs) in the standard map serve as the motivation background for this paper [29][30][31][32][33][34][35]. Yet, despite the simple dynamics, interesting applications for this system can be found in dynamic stability in human performance [36], vibration waves in a nanometric-sized mechanical contact system [37], granular materials [38], experimental devices concerning normal coefficient of restitution [39], mechanical vibrations [40,41], anomalous transport and diffusion [42], thermodynamics [43], crisis between chaotic attractors [44], and chaos control [45], among others [46,47].…”

Section: Introductionmentioning

confidence: 99%

Transition from normal to ballistic diffusion in a one-dimensional impact system

et al. 2018

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We characterize a transition from normal to ballistic diffusion in a bouncing ball dynamics. The system is composed of a particle, or an ensemble of noninteracting particles, experiencing elastic collisions with a heavy and periodically moving wall under the influence of a constant gravitational field. The dynamics lead to a mixed phase space where chaotic orbits have a free path to move along the velocity axis, presenting a normal diffusion behavior. Depending on the control parameter, one can observe the presence of featured resonances, known as accelerator modes, that lead to a ballistic growth of velocity. Through statistical and numerical analysis of the velocity of the particle, we are able to characterize a transition between the two regimes, where transport properties were used to characterize the scenario of the ballistic regime. Also, in an analysis of the probability of an orbit to reach an accelerator mode as a function of the velocity, we observe a competition between the normal and ballistic transport in the midrange velocity.

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“…[11][12][13][14] On the other hand, an alternative model was proposed by Pustylnikov 15,16 which consists of a classical particle bouncing in a vertical moving platform, having, instead of a fixed wall, an constant gravitational field working as returning mechanism. [17][18][19][20][21][22] For such a system and for specific combinations of both control parameters and initial conditions, the phenomenon of unlimited energy growth can be observed due to the loss of correlation between two collisions.…”

Section: Introductionmentioning

confidence: 99%

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

Oliveira¹,

Leonel²

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Some dynamical properties for a time dependent Lorentz gas considering both the dissipative and non dissipative dynamics are studied. The model is described by using a four-dimensional nonlinear mapping. For the conservative dynamics, scaling laws are obtained for the behavior of the average velocity for an ensemble of non interacting particles and the unlimited energy growth is confirmed. For the dissipative case, four different kinds of damping forces are considered namely: (i) restitution coefficient which makes the particle experiences a loss of energy upon collisions; and in-flight dissipation given by (ii) F ¼ ÀgV 2 ; (iii) F ¼ ÀgV l with l 6 ¼ 1 and l 6 ¼ 2 and; (iv) F ¼ ÀgV, where g is the dissipation parameter. Extensive numerical simulations were made and our results confirm that the unlimited energy growth, observed for the conservative dynamics, is suppressed for the dissipative case. The behaviour of the average velocity is described using scaling arguments and classes of universalities are defined. We revisit the problem of non-interacting particles in a time dependent Lorentz gas. We describe the model by using a four dimensional nonlinear map. As it is known, the phase space of the Lorentz gas with static scatterers is fully chaotic and the velocity of the particle is constant. However, when a time dependent perturbation is introduced to the boundary, the energy is no longer conserved and the unlimited energy growth is observed for the conservative case. On the other hand, our results show that when dissipation is introduced into the system, either by collisional dissipation or dissipation during the flight, the unlimited energy growth is no longer observed. Depending on the strength of the dissipation and considering short time, the average velocity can either grows and reaches a constant plateau or decays until the particle reaches the state of rest. For the cases, when the dynamics does not stop between the scatterers, the behaviour of the average velocity is described by using scaling arguments and once the scaling exponents are known, classes of universalities are defined.

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The dynamics of a bouncing ball with a sinusoidally vibrating table revisited

Cited by 102 publications

References 16 publications

Simple Model of Bouncing Ball Dynamics

Simple Model of Bouncing Ball Dynamics

Transition from normal to ballistic diffusion in a one-dimensional impact system

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

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