Dynamics of a dissipative, inelastic gravitational billiard

Abstract

The seminal physical model for investigating formulations of nonlinear dynamics is the billiard. Gravitational billiards provide an experimentally accessible arena for their investigation. We present a mathematical model that captures the essential dynamics required for describing the motion of a realistic billiard for arbitrary boundaries, where we include rotational effects and additional forms of energy dissipation. Simulations of the model are applied to parabolic, wedge and hyperbolic billiards that are driven sinusoidally. The simulations demonstrate that the parabola has stable, periodic motion, while the wedge and hyperbola (at high driving frequencies) appear chaotic. The hyperbola, at low driving frequencies, behaves similarly to the parabola; i.e., has regular motion. Direct comparisons are made between the model's predictions and previously published experimental data. The representation of the coefficient of restitution employed in the model resulted in good agreement with the experimental data for all boundary shapes investigated. It is shown that the data can be successfully modeled with a simple set of parameters without an assumption of exotic energy dependence.

A Three Dimensional Gravitational Billiard in a Cone

“…where f (τ n+1 , r n , v nr ) = 0 is defined by the left hand side of (10). The computations are straightforward but tedious due to the fact that v n θ depends on r n and v nr in the energy expression (14).…”

“…In addition quantum versions of these systems have also been studied by Waalkens et al [5] among others. While these classical billiards are optimal for analytical study, more experimentally approachable models accounting for the Earth's gravitational field, called gravitational billiards, have also been widely studied [6][7][8][9][10][11][12][13][14][15]. In [6], the wedge billiard, consisting of a particle falling between two symmetric linear boundaries of angle 2θ, was shown by Lehtihet and Miller to exhibit the full range of possible behavior in Hamiltonian systems with two degrees of freedom.…”

“…The gravitational bouncer was introduced as a variant of the well-known Fermi-Ulam model [26,27], and has been studied for several types of driving motions including sinusoidal [20][21][22] and piecewise linear [23][24][25]. To model the Feldt experiments, two-dimensional driven gravitational billiards were studied numerically in [14,28]; in the former theoretical study rotational effects were ignored, while in the latter rotational effects were included in the model, making analytical computations e.g., periodic orbits difficult. However, in the work of Hartl et al [14], the numerical simulations supported the experimental results of [18], except in secondary quantities derived from the data, such as the tangential velocity of the particle after collisions.…”

Ball bouncing down rounded edge stairs: chaotic but tricky

“…The motion was found to be nonchaotic (typically quasi-periodic), therefore, here we turn to the question of whether by keeping the framework of gravitational billiards, that is point-like non-rotating ball with negligible air drag in a gravitational field (see e.g. [4][5][6]), a mere change in the geometry, namely the rounding of the edges (resulting in what we call rounded stairs in short), is sufficient to generate chaos. Intuitively, this is a natural expectation, and the authors of [2] conjectured indeed that the dynamics will be chaotic since the curvature at the edge will act as a magnifying mirror, which scatters parallel incoming beams (while the rectangular case contained only plane mirror-like surfaces).…”

Chaotic or just complicated? Ball bouncing down the stairs

“…[20][21][22][23]), which are typically considered without any energy loss (k = 1), because otherwise with the lack of driving force, any motion would ultimately stop. The literature is much more restricted for billiards in the presence of gravity, some of which, such as wedgetype billiards, are known to support chaotic dynamics [24,25].…”