COMPUTER SIMULATION OF THE
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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.
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  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 , Marcos and Massoud , Rachner and Jungk .
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  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.
d2x/dt2 = (f(t) - Ddx/dt)/M