On the stopping time of a bouncing ball (original) (raw)
Related papers
2009
The dynamics of a bouncing ball undergoing repeated inelastic impacts with a table oscillating vertically in a sinusoidal fashion is studied using Newtonian mechanics and general relativistic mechanics. An exact mapping describes the bouncing ball dynamics in each theory. We show that, contrary to conventional expectation, the trajectories predicted by Newtonian mechanics and general relativistic mechanics from the same parameters and initial conditions for the ball bouncing at low speed in a weak gravitational field can rapidly disagree completely. The bouncing ball system could be realized experimentally to test which of the two different predicted trajectories is correct.
We study a simple model of a bouncing ball that takes explicitely into account the elastic deformability of the body and the energy dissipation due to internal friction. We show that this model is not subject to the problem of inelastic collapse, that is, it does not allow an infinite number of impacts in a finite time. We compute asymptotic expressions for the time of flight and for the impact velocity. We also prove that contacts with zero velocity of the lower end of the ball are possible, but non-generic. Finally, we compare our findings with other models and laboratory experiments.
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.
Simple models of bouncing ball dynamics and their comparison
2010
Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Several simple models of table motion are studied and compared. Dependence of displacement of the table on time, approximating sinusoidal motion and making analytical computations possible, is assumed as quadratic and cubic functions of time, respectively.
arXiv (Cornell University), 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.
Feedback Control of Impact Dynamics: the Bouncing Ball Revisited
Proceedings of the 45th IEEE Conference on Decision and Control, 2006
We study the the design of a tracking controller for the popular bouncing ball model: the continuous-time actuation of a table is used to control the impacts of the table with a bouncing ball. The proposed control law uses the impact times as the sole feedback information. We show that the acceleration of the table at impact plays no role in the stability analysis but is an important parameter for the robustness of the feedback system to model uncertainty, in particular to the uncertainty on the coefficient of restitution.
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.
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.
The impact of a soft contractile body on a hard support is described by fields of short range forces. Besides repulsion these forces are able to describe also friction, damping and adhesion allowing the body to have complex motions which look rather realistic. The contractility is used to make the body look like a living body with some basic locomotion capabilities. The simulated motion, like jumping or crawling, is driven either by a contraction or by the corresponding force. Although only affine motions are allowed, the model arises from a general theory of remodeling in finite elasticity and shows also creep and plastic deformations. The body is made of a viscous incompressible Mooney-Rivlin material.