The dynamics of a bouncing ball with a sinusoidally vibrating table revisited (original) (raw)
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Bouncing ball problem: Stability of the periodic modes
Physical Review E, 2009
Exploring all its ramifications, we give an overview of the simple yet fundamental bouncing ball problem, which consists of a ball bouncing vertically on a sinusoidally vibrating table under the action of gravity. The dynamics is modeled on the basis of a discrete map of difference equations, which numerically solved fully reveals a rich variety of nonlinear behaviors, encompassing irregular nonperiodic orbits, subharmonic and chaotic motions, chattering mechanisms, and also unbounded nonperiodic orbits. For periodic motions, the corresponding conditions for stability and bifurcation are determined from analytical considerations of a reduced map. Through numerical examples, it is shown that a slight change in the initial conditions makes the ball motion switch from periodic to chaotic orbits bounded by a velocity strip v = Ϯ⌫/ ͑1−͒, where ⌫ is the nondimensionalized shaking acceleration and the coefficient of restitution which quantifies the amount of energy lost in the ball-table collision.
An analytical and numerical study of chaotic dynamics in a simple bouncing ball model
Acta Mechanica Sinica, 2011
We study dynamics of a ball moving in gravitational field and colliding with a moving table. 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 moves down with another constant velocity. The Poincaré map, describing evolution from an impact to the next impact, is derived and scenarios of transition to chaotic dynamics are investigated analytically.
Nonlinear Dynamics, 2011
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.
Bifurcations and chaos for the quasiperiodic bouncing ball
Physical Review E, 1997
We investigate the influence of a second frequency on the classical periodic bouncing-ball problem, and call it the quasiperiodic bouncing ball. We indicate how to compute the Lyapunov exponent for implicit maps and confirm the presence of chaos for the periodic bouncing ball. We have numerically found a series of nontrivial bifurcations for the quasiperiodic bouncing ball. We have also found several cases of nonperiodic attractors with negative Lyapunov exponents. ͓S1063-651X͑97͒02710-4͔
Simple Model of Bouncing Ball Dynamics
Differential Equations and Dynamical Systems, 2012
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.
Journal of the Physical Society of Japan, 2002
Motivated by bouncing motion of an inelastic particle on a vibrating board, a simple two-dimensional map is constructed and its behavior is studied numerically. In addition to the typical route to chaos through a periodic doubling bifurcation, we found peculiar behavior in the parameter region where two stable periodic attractors coexist. A typical orbit in the region goes through chaotic motion for an extended transient period before it converges into one of the two periodic attractors. The basin structure in this parameter region is almost riddling and the fractal dimension of the basin boundary is close to two, i.e., the dimension of the phase space.
Dynamical properties of a non-autonomous bouncing ball model forced by non-harmonic excitation
Mathematical Methods in the Applied Sciences, 2016
The main aim of the paper is to research dynamic properties of a mechanical system consisting of a ball jumping between a movable baseplate and a fixed upper stop. The model is constructed with one degree of freedom in the mechanical oscillating part. The ball movement is generated by the gravity force and non-harmonic oscillation of the baseplate in the vertical direction. The impact forces acting between the ball and plate and the stop are described by the nonlinear Hertz contact law. The ball motion is then governed by a set of two nonlinear ordinary differential equations. To perform their solving, the Runge-Kutta method of the fourth order with adaptable time step was applied. As the main result, it is shown that the systems exhibit regular, irregular, and chaotic pattern for different choices of parameters using the newly introduced 0-1 test for chaos, detecting bifurcation diagram, and researching Fourier spectra.