Group 3

Typical position of the ball (blue) and plate (black) versus time for c=3.

The appropriately-named bouncing inelastic ball experiment is composed of a perfectly inelastic ball bouncing on an oscillating plate. It is a simple yet elegant system that exhibits striking nonlinear behavior<ref>A. Mehta and J. M. Luck, Physical Review Letters 65, 393 (1990)</ref>.

Theory

The motion of a bouncing ball on a simple cosine forced plate has been thoroughly investigated<ref>T. Gilet, N. Vandewalle, and S. Dorbolo, Physical Review E 79, 055201(R) (2009)</ref>, so we aim to look at a forcing function composed of a superposition of two sine waves with integer-multiple frequencies. The motion of our plate is as follows:

<math> \begin{equation} x_p(t) = A[\sin(\omega t) + \sin(c \omega t)] \end{equation} </math>

The ball can take off from the plate when the acceleration of the plate is less than negative g, the acceleration due to gravity: <math> \begin{align} \ddot{x}_p(t) = -A[\omega^2\sin(\omega t) + c^2 \omega^2\sin(c \omega t)] &\leq -g \notag\\ -\frac{A\omega^2}{g} [\sin(\omega t) + c^2 \sin(c \omega t)] &\leq -1 \end{align} </math>

In this way, the motion of the ball from initial takeoff at <math>t_0</math> is:

<math> \begin{align} x_b(t) = &x_p(t_0) - x_p(t) + \dot{x}_p(t_0)(t-t_0) - 1/2 g(t-t_0)^2 \notag\\ x_b(t) = &A[\sin(\omega t_0)+ \sin(c \omega t_0) - \sin(\omega t) - \sin(c \omega t)] + A[\omega \cos(\omega t_0) + c \omega \cos(c \omega t_0)](t-t_0) - 1/2 g(t-t_0)^2 \notag \end{align} </math>

<math> \Gamma, \varphi, \chi</math> regime

The above functions can be made dimensionless via the introduction of a characteristic time <math>1/\omega</math> and a characteristic length <math>g/\omega^2</math>. We can thus define a phase <math>\varphi = \omega t</math>, a reduced acceleration and one of our control parameters <math>\Gamma = A\omega^2/g</math>, and a generalized position <math>\chi = A\omega^2/g</math>. The other control parameter is the ratio of the frequencies <math>c</math>.

The forcing of the plate is now given by:

<math> \begin{equation} \chi_p(\varphi) = \Gamma(\sin(\varphi) + \sin(c \varphi)) \end{equation} </math>

The position of the ball relative to the plate is

<math> \begin{align} \chi_b(\varphi) = &\Gamma[\sin(\varphi_0)+ \sin(c \varphi_0) - \sin(\varphi) - \sin(c \varphi)] + \dots \\ &\Gamma[\cos(\varphi_0) + c\cos(c \varphi_0)](\varphi - \varphi_0) - \dots \notag \\ &1/2(\varphi-\varphi_0)^2. \notag \end{align} </math>

and the ball takes off when

<math> \begin{equation} \Gamma [\sin(\varphi) + c^2 \sin(c \varphi)] \leq -1 \end{equation} </math>

Period Doubling and Chaos

The flight time is defined as <math>T = \varphi_i - \varphi_0</math>, and can be found by solving the following equation numerically:

<math> \begin{align} F \equiv &\Gamma [\sin(\varphi_0)(1-\sin T) + \sin(c\varphi_0)(1-\sin cT) - \cos T \cos(\varphi_0) - \cos(cT)\cos(c\varphi_0)] - \dots \\

       &\Gamma[\cos(\varphi_0) + c\cos(c\varphi_0)]T - 1/2 T^2 = 0

\end{align} </math>

The ball is able to land in two distinct regions. There is a 'bouncing' region, where the acceleration of the plate is such that the ball will immediately take off again, with an initial velocity equal to the instantaneous velocity of the plate at the time of impact. The other region is the 'sticking' region, where the ball is not able take off, and instead rests on the plate until the plate's acceleration is such that is can take off again. This region is a sort of resetting region, as the ball immediately forgets any history it had prior to landing. The sticking region will mark the boundary between periodic cycles.

By varying the control parameter <math>\Gamma</math> we aim to construct bifurcation diagrams for the cases <math>c=1, c=2</math> and <math>c=3</math>.

Experiment

The experimental setup involves a horizontal shaking platform driven vertically according to the forcing function inputted by an oscilloscope or computer. A small, inelastic ball is placed on the plate and it's motion is recorded by a high speed camera. The camera data is then transferred into position data for both the ball and the plate to be compared with the predictions made by our computational model. Below is a list of required components.

Parts list

1. oscilloscope or computer capable of outputting different forcing functions
2. shaker plate
3. completely inelastic hacky-sack type ball
4. high speed camera with computer interface
5. tracking software

Results

To be found.

References

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