**COMPUTER SIMULATION OF THE
OSCILLON PHENOMENON**

Universidad Panamericana

Augusto Rodin 498, 03910 Mexico City

fax (525) 563 3747

http://www.raczynski.com

stanracz@prodigy.net.mx

**SUMMARY**

In many industrial processes it is necessary to design devices that handle or process
granulated materials or “granular media”. The dynamics of such media is rather difficult to
model and simulate. In fact, the granular media problem leads to a multibody problem with
millions of particles. The multiple collisions and friction between particles make the model
extremely non-linear and difficult to simulate. This paper describes only a simplified model
of the newly discovered oscillon phenomenon. However, it could contribute to a better
understanding of the dynamics of granular media. In particular, it is shown that while
externally excited, the granular media can reveal at least two stable oscillation regimes, that
can coexist one near the other and survive for a long time.

**INTRODUCTION**

Many industrial processes handle granulated materials like sand, granulated plastic or
grain. The analysis of the dynamics of such granular media (GM) is a difficult task. It is
known that it can not be treated as a liquid and its mathematical description is rather
complicated. Computer simulation can help to point out some interesting properties. One of
the most interesting phenomena observed in GM is the formation of oscillons, being local
changes in the state of externally excited GM.

In Scientific American, November 1996, Madhusree Mukerjee [3] defines oscillon as “a
pile of tiny brass balls that jiggles up and down and joins with other piles to form patterns”.
The phenomenon was discovered by Paul B. Umbanhowar, F. Melo and H.L. Swinney
[5,6,8,9] at the University of Texas in Austin. They vibrated a tray with brass balls of 0.1
millimeter in radius up and down with frequencies between 10 and 100 cycles per second. In
those articles it was pointed out that serious theoretic difficulties appear while dealing with a
sand-like GM. Though the media resembles liquid when fed with energy, its dynamics is
quite different and somewhat mysterious. Computer simulation can help to prove some
properties of the GM. Possible applications of the GM theory in material handling equipment
are well known. The knowledge of the dynamical properties of the GM is essential, for
example, while designing oscillating conveyors. Important advances of the GM theory has
been made mainly by those who work on problems related to such kind of equipment.
Consult Gaberson [2], Marcos and Massoud [4], Rachner and Jungk [7].

**SIMULATING OSCILLONS**

The following experiment is a simple two-dimensional simulation of a set of moving
balls. The aim of this task is to point out that the GM can reach at least to steady vibration
regimes, when excited by a vibrating tray. The reason for this assessment is very simple: the
friction between the particles introduces damping strong enough to turn off oscillations
when the particles are close each other. However, if the particles are in faster movement,
then the elastic collisions are more frequent than friction-related clustering. As a result, in
the regions when the vibration is strong, the damping drops and the vibrations does not
disappear (supposing the same external excitation). This means that with the same excitation
one could observe a slow damped movement as well as fast vibrations in other regions. The
problem is to show that the two steady states can coexist in the same set of particles and that
the excitations may be stable, that is, may survive for a long time without changing their
shape.

To describe what exactly happens in a GM one has to use a complicated non-linear
equation of motion of each particle. To simulate a real oscillon is quite impossible for the
dimensionality of the problem. However, a simple two-dimensional model can be built and
run successfully even on a PC. The following experiment merely shows the possibility of the
coexistence of the two steady states. The model is very simplified and rather abstract. The
rules of movement are as follows. A set of circular particles is subject to gravity
acceleration. The collisions between particles are modeled as movements caused by elastic
forces that appear when two particles touch each other. The forces generated during
spherical particle collisions obey the rules given by the well known Hertz problem. In
general, it is known that the force is proportional to x3/2, x being the relative displacement of
the centers. The damping coefficient of the movement (energy dissipation) depends on the
actual situation. If the particle is in touch with more than one other particles, then the energy
dissipation grows considerably. This can be explained by the fact that the friction between
particles damps the movement when the particles form clusters, like the sand on a beach.
The model was simulated using an object-oriented simulation tool (PASION simulation
system).

Each particle was generated as an instant of a BALL process, and activated. 185 particles
were launched, each one moving according to the forces it receives. In other words, each
particle resolves its own movement equation, integrating it with certain integration step. This
approach has several advantages, comparing with the integration of one global system of
370 differential equations of first order. First, the program is very simple and permits to
include any other events (discrete or continuous), running concurrently in the same program.
Second, the particles are separate objects, each one equipped with its own parameters. The
time step for integration of the movement equation can be different for each particle. For
example, if a particle moves alone and has no near neighbors, then the step can be greater
than that of a particle which is in contact or approaches other ones. This can accelerate the
simulation. A disadvantage of such object-oriented simulation compared with one Ordinary
Differential Equations (ODE) system is that the ODE model may run faster, when an
adequate numerical method is used. Note, however, that in this case we need 370 non-linear
and rigid differential equations. Recall that the collisions between rigid bodies lead to rigid
systems of equations that may provoke serious numerical difficulties. Of course, the object-
oriented approach is not a magic remedy for such difficulties, and the collisions must be
modeled as if they occur between particles more elastic than brass balls. Looking at the
movement of the simulated GM it can be seen, however, that the particles behave like
colliding brass balls. At any rate the model and the results are rather qualitative and are not
related to any real physical parameters.

It is not an easy task to simulate a stable oscillon. Its stability depends on all model
parameters, like forces, damping coefficients, tray frequency, amplitude etc. One simulation
takes about half hour (100 Mhz CPU was used), if one wants to verify if the oscillon is
stable. If the tray amplitude is low, then the oscillon dies. If the excitation is to big, it grows
and the simulation terminates in totally chaotic ball movements (the model switches to the
strong oscillation regime). After one week of work, the first stable oscillon appeared on the
screen. It had the diameter of about 20 times the ball diameter and drifted very slowly. What
was essential in this simulation is that the oscillon remained almost unchanged for a long
time. Two separate oscillons can also be simulated, though they are not so stable and after
some time join each other. To simulate multiple oscillons, smaller balls are needed and this
makes the simulation slow, because the number of balls must be greater.

Sometimes in such a simulation an oscillon appears spontaneously, but it is more
probable to obtain it introducing an external disturbance (Mukerjee [3] says that an oscillon
can be formed by touching the vibrating GM with a pencil). The simulation scenario was as
follows. First, the tray does not vibrate and the balls appear with some initial elevation and
fall down (phase 1). Then they form a stable layer over the tray. Next, the tray begins to
vibrate (phase 2). It can be seen that the balls vibrate too, but the vibration regime can not
switch to the strong one. After some time a short disturbance is applied in certain region.
The balls affected by the disturbance “explode” and begin to move fast, colliding with each
other (phase 3). This strong vibrations became stable, i.e. the strongly vibrating region
remains unchanged for a long time. The other balls also vibrate, but form a stable cluster.

Figure 1. Oscillon - view of a simulation screen.

Figure 1 shows the situation on the tray. The balls in the oscillon jump up and fall. If a ball falls out of the oscillon limits, it enters in a cluster and stabilize. On the other hand, some balls near to the oscillon leave the cluster and begin to vibrate strongly. Figure 2 shows the ball trajectories registered over certain time interval.

Figure 2. Ball trajectories of the oscillon of figure 1.

Figure 3. Trajectories in two separate oscillons.

Figure 4. The average elevation of balls within an oscillon as a function of time.

Each particle moved according to its movement equation

d2x/dt2 = (f(t) - Ddx/dt)/M

It is clear that the model does not pretend to simulate a real 3-dimensional GM, but perhaps its qualitative properties may be similar to a real GM. In particular, the ability to form oscillons confirms such similarity. The simulation program (source and EXE) is available from the author on request. It should be emphasized that the model is highly sensitive to all its parameters, and any other parameter configuration may require many trials to obtain stable oscillons.

PASION is not the only simulation tool that can be applied to simulate the dynamics of multibody systems. Perhaps the simulation described here could be coded in any other object-oriented language, like Simula, ISIM, ACSL, CSSL, or general purpose languages like C++ etc. (see the Directory of Simulation Software [1] for more than a hundred of other packages). PASION was selected because it is commonly used in our Center of the McLeod Institute for Simulation Sciences at the Universidad Panamericana, and because it is a good object-oriented simulation tool. Briefly speaking, it offers a simple and clear program structure, divided into process declarations. Each process contains a sequence of events. The events can be related to model time (scheduled), or to model state (state events). Some model processes can be continuous and others discrete, running concurrently in the same simulation program. At run time objects are created as instances of the process declarations. They may be activated, disactivated, suspended or erased. Each active object can interact with any other according to its events. PASION system includes source code generators for queuing models, models described by signal flow diagrams, bond graphs, differential equations, transfer functions and any combination of the above. The efficient clock mechanism makes the time management transparent and partially hidden from the user. The discrete part of the package also contains an animator module, applied mainly to queuing models. For more detail consult the pages listed in PASION LINKS section

This, rather qualitative and behavioral simulation can point out some interesting properties of the GM. PASION of course, is not the only tool that can be used in this task. If you have a lot of time you can try to simulate oscillons using more traditional methods, like the huge set of differential equations. Perhaps using a parallel supercomputer one could make 3D simulations of the GM and more realistic oscillon images. A possible applications of similar models and simulations can be made in any industrial engineering problems that involve GM handling, like oscillating conveyors and granular material processing.

[1] "Directory of Simulation Software", The Society for Computer Simulation, San Diego, CA. Appears each year.

[2] Gaberson H.A, 1972, "Particle Motion on Oscillating Conveyors", Transaction of the ASME, Journal of Engineering for Industry, February 1972, pp. 50-63.

[3] Madhusree Mukerjee, 1996, "Science with Brass", Scientific American, November 1996, pp.19-21.

[4] Marcos W.A. and Massoud M.F., 1969, "On the Design of Oscillating Conveyors", Transaction of the ASME, Journal of Engineering for Industry, May 1969, pp.353-356.

[5] Melo F., Umbanhowar P.B, Swinney H.L., 1994, "Transistion to parametric wave patterns in a verically oscillated granular layer", Phys. Rev. Lett., 72, 172-175.

[6] Melo F., Umbanhowar P.B, Swinney H.L., 1995, "Hexagons, kinks and disorder in oscillated granular layers", Phys. Rev. Lett., 75, 3838-3841.

[7] Rachner H.G. and Jungk L., 1967, "Material Transport on Oscillating Conveyors", PhD thesis, Massachussetts Institute of Technology.

[8] Umbanhowar P.B, 1996, "Pattern formation in vibrated granular layer", Paper Q5.02, APS March Meeting, St. Luis, MO.

[9] Umbanhowar P.B, Melo F., Swinney H.L., 1996, "Localized excitations in a vertically vibrated granular layer", Nature 382, 793-796.

PASION links:

http://www.raczynski.com

http://www.raczynski.com/pn/pn.htm

http://www.raczynski.com/pn/sumduz.htm

http://www.raczynski.com/art/artyk1.html