

A simple variation of Fermi's model will be used to show the effect of
collision process in the rich dynamical response of impact systems.
A dynamic schemata of the model is shown on top left of this
page,
which shows a ball falling freely under the force of gravity and "bouncing" of
a "periodically driven" table. Well, it is not a 'real' simulation but it does
create a visual illusion of the simple model described above!! It is not a
realistic simulation in the sense that the motion of the ball is not determined
by Newton's law and the collision process is not modelled by any rule. Having
said that now if we assume that we do simulate a realistic model, we want to
find out how does the motion of the ball depend upon the system parameters.
Consider the motion of a ball bouncing on a periodically vibrating table.
To keep the matters simple, we will
assume that the ball's motion is confined to the vertical direction and that,
between impacts, the ball's height is determined by Newton's law for a
freely falling particle under the influence of gravity. Thus, between the
impacts, the position and the velocity of the ball at any given time is given by
x(t) = x_{0} + v*t
v(t) = v_{0}  g*t
where x_{0} and v_{0} are the initial position and the velocity
and 'g' denotes the acceleration due to gravity.
The collision process is
modelled using coefficient of restitution rule.
It is assumed that the table's mass is
much greater than the ball's mass and the impact between
the ball and the table is instantaneous. The implication of these assumptions
is that the table's motion is unaffected by the collisions.
Assuming that motion of the table is igiven by a sinusoidal form
The equations of motion are reset, at the time of impact by the
following expression
v^{+} = K*(u  v^{}) + u
where u is the velocity of the table at the instant
the collision occurred and the parameter K called the coefficient
of restitution, is a measure
of amount of energy lost in the process of collision (actually 1K^{2
}). The value of K ranges between 0 and 1. Values of K less
than 1 imply inelastic collision (dissipative system) and K = 1 implies
elastic collision (conservative system). Note, in either case there is
a nonlinear force that
is applied to the ball when it hits the table. At impact, the ball's
velocity suddenly reverses from the downward to the upward direction
which appears as a sharp discontinuity in the phase space describing
the motion of the ball (position velocity plot).
It turns out simulating this simple model one observes, for a wide range of the
drive frequency (a system parameter), keeping the other parameter fixed, this
simple dynamical system shows chaos!!
This is surprising, given the fact that if we assume the table to be held
fixed the ball bouncing of the table would either eventually come to rest
on the table (if energy is lost as a result of collision  dissipative system)
or the ball oscillates with the same amplitude (if there is no loss of energy
as a result of the collision  conservative system).
Of the several equivalent ways that exists to represent and study the dynamics
of bouncing ball system, we will show the motion of the ball by measuring
the time between impacts and see how the eventual behavior of the ball
changes as we change one of the system parameters. All plots shown here
will be for the case of inelastic collisions, note for a driven bouncing
ball system with elastic collision, the motion is unbounded and thus
dismissed for being uninteresting!
So finally, here it is!! The chaotic attractor of the bouncing ball system
for the A = 0.012, K = 0.5 and drive frequency 2*PI*6 rads/sec, reconstructed
using the time between impacts. The attractor is embedded in 3D space. The
following pictures are for same parameter values as above except that
the views are different.

Figure 1: In all figures time between impact is used to reconstruct the
attractor with the attractor embedded in 3D space. The parameter values
chosen are A=0.012, K = 0.5 and drive frequency = 2*PI*6.0 rad/sec. The
plot is t_{i2}, t_{i1}, and t_{i} for X, Y and
Z axis, respectively. Thus, the plot essentially is a return map in 3D and
the intersection of the body diagonal passing through the origin with any point
on the chaotic attractor gives the location of the embedded unstable fixedpoint
attractor of the map. The view used for plotting the above attractor are
top left (0 0), top right (45 45), bottom left (90 180) and bottom right
(45 255). The Xaxis, before the rotation, is the horizontal axis
pointing to the right, the Yaxis the vertical axis pointing vertically up and
the Zaxis is the axis pointing out of the plane (towards you!). Then the
first angle represents the rotation about the Xaxis and the second angle
represents the rotation about the new Zaxis.

Note the embedding of the attractor in 3D space was important from the
point of view of controlling the unstable fixed point embedded in the
chaotic attractor by precluding the false nearest neighbors when the control
constants are calculated using the linear dynamics around the fixed point.
Following is the bifurcation plot for the bouncing ball system as the drive
frequency is changed from 2*PI*5.5 rad/sec. to 2*PI*6.1 rad/sec. The plots show
the time between impacts and the relative velocity of the ball with respect to
the velocity of the table at the time of the impact. It is clear from the
plot that for the parameter values chosen (K=0.5 and A=0.012) the bouncing
ball goes through period doubling and eventually becomes chaotic. The chaotic
invariant set shown in figure 1 goes through crisis at drive frqeuency
sightly above 2*PI*6.0 and it destroys the chaotic attractor. Beyond this
crisis point the system either shows long transient chaos or the orbit
follows the shadow of the chaotic attractor before it eventually settles down
to a sticking solution.

Figure 2: Bifurcation plot for bouncing ball system as a function of the
drive frequency. Top: impact interval against the drive frequency.
Below: Relative velocity against the drive frequency.



STICKING SOLUTIONS  GRAZING IMPACTS 

As shown for the case of impact oscillator
system, the grazing collision
of the orbit leads to extreme stretching and contraction in region of the phase
space near the colliding boundaries and leads to destruction of the orbit
that grazes the wall and is sometime followed by transient chaos or
intermittency. The lower plot in figure 2 clearly shows the effect of grazing
collisions (when the relative velocity is zero just before the impact).

Figure 3: Reconstructing the chaotic attractor (shown in green for drive
frequency 2*PI*6.0) using the impact time interval, just before the
attractor goes through crisis. The points in red show the shadowing of the
chaotic attractor by the orbits that goes through grazing collisions. The
drive frequency for points plotted in green is 2*PI*6.03987. The range of
impact time interval is from 0.0 to 2.2. All other system parameters are
set as before.

All figures on this page were produced from this simple
Ccode that generates
the data file and could either be piped through my
data to ps converter (dps) program which outputs a postscript file or
use any of the
plotting program to plot the data. Once in postscript format, use the following
command line to convert the file into gif format.
gs sDEVICE=ppmraw sOutputFile= sNOPAUSE q infile.ps c showpage c quit  pnmcrop  pnmmargin white 10  ppmtogif > outfile.gif
The file infile.ps is the ps file supplied by you. You need to have Pbmplus
utilities installed for this commandline to work. Well all said and done
there is an XWindow and
DOS version of the program
that plots the data as it is
generated, allows to view the attractor in 3D at any view angle. The program
also implements the recursive proportional control strategy in conjunction
with adaptive learning algorithm to stabilize the unstable periodic orbit
of the impact system. The program can be used as template to simulate any
system modelled by ordinary differential equations or maps.
The programs are set up to control a sinusoidally driven nonlinear
pendulum.
 [1]

C. N. Bapat, S. Sankar, and N. Popplewell,
Repeated impacts on a sinusoidally vibrating table reappraised, J. Sound Vib.
108 (1), 99115 (1986).
 [2]

Nicholas B. Tufillaro, Jeremiah Reilly, and Tyler Abbott,
An experimental approach to nonlinear dynamics and chaos, AddisonWesley, 1992.
 [3]

J. Thomas, M. A. Rhode, and R. W. Rollins,
Chaos in an impact oscillator: experiment, simulation, and control,
presented at Dynamics Days Arizona, 16th Annual International
Conference, Scottsdale, AZ, January 811, 1997.
