| |
ISSN 1320-0682 | |||
| Volume 02 | April 1995 | |||
Han Huang, Jie Pan and Paul G. McCormick
Department of Mechanical and Material Engineering
University of Western Australia
Nedlands, WA 6009, Australia
Email: hhuang@shiralee.mech.uwa.edu.au
The vibratory ball mill has been used both as a device for mechanical alloying [1] and as a tool to investigate the effect of system dynamics on the milling process [2]. Similar to the bouncing ball system, the vibratory ball mill exhibits a rich complexity of dynamic behaviour, which can be characterised as chaotic phenomena. The chaotic characteristics of the bouncing ball system have been studied for many years. The period doubling route to chaos [3, 4], stability and boundaries of the period orbits [5, 6], suppression of period doubling induced by the near-resonance perturbations [7, 8] and sensitive dependence of chaotic motion on the initial conditions [3] are characteristic behaviour of the system. It has been shown that the coefficient of restitution, e, for collision events determines the nature of the observed dynamics [3, 5]. In engineering applications, e is a function of many parameters, such as the mass, size, and material properties of colliding objects; thus, it is of particular value to study the effect of e on the system dynamics. In this paper, the effect of the coefficient of restitution on the chaotic motion is systematically numerically investigated in terms of strange attractors and the corresponding correlation dimensions [9]. The chaotic phenomena occurring in an experimental vibratory ball milling system is analysed and compared with the simulation results. The effects of perturbing factors in the ball milling system on the observed chaotic behaviour are discussed with the aid of numerical analysis.
A simple physical model of the bouncing ball system is illustrated in Figure 1(a). The platform is sinusoidally excited in the vertical direction, and its motion is described by
where A and
are the amplitude and angular frequency
of the platform, respectively, and
is the initial
phase of the platform. The ball starts to depart the platform
when the inertial force,
, acting on it is
sufficiently large to overcome the earth's gravitational force,
mg, where m is the mass of the ball. The starting phase
( 
) of the ball is
and its starting velocity is equal to the platform velocity at the phase. Once the ball starts to move, its trajectory is a projectile
where 
is the position at the kth impact,
is the
velocity immediately after kth collision, and
is the
instance of kth collision. The next collision occurs if the
difference in position between the ball and platform is zero
Assuming that the collision does not influence the motion of the platform, the change in the ball's velocity at collision is then calculated from the impact relation
where 
is the velocity of the platform at the kth
collision, and
and
are the velocities of the ball
immediately before and after the kth collision, respectively.
Therefore, the ball's trajectory is determined by specifying the
phase of collision and the velocity of the ball immediately
before and after collision.
Figure 1: (a) The bouncing ball system;
(b) The vibratory ball mill, (1) platform, (2)
container, (3) ball and (4) accelerometer.
Computer simulations were performed based on
Equations (1) to (5). Initially, the ball was resting on the platform. Its
starting position for bouncing was determined by Equation (2)
and the corresponding velocity was equal to the platform
velocity at the starting position. The time of next collision
was obtained by solving Equation (4) numerically using a time
step interval of 10 microseconds. The next starting velocity of
the ball was determined by Equation (5). The process was
repeated until the completion of 4000 collisions, which ensured
the system reached a steady state. The time intervals between
two consecutive collisions were recorded in order for mapping
out the strange attractors. The milling experiment was carried
out in a single axis vibratory ball mill. The schematic of the
mill is shown in Figure 1(b). During the experiment, a grinding
ball, 50 mm in diameter, and 15 grams of 304 stainless
steel powder were used. The vibratory mill is actually a
bouncing ball system with high damping caused by milled powder.
The value of
used for simulation was estimated
from an approximate free falling experiment. The acceleration of
the milling system was monitored by the accelerometer and
transmitted to a digital computer for identifying the occurrence
and position of collisions and calculating the time intervals.
It is well known that for a lightly damped bouncing ball system, the bifurcation diagram shows the classic period doubling route to chaos [3]. The bifurcation diagram for the vibratory ball milling system generated by the simulation program is shown in Figure 2. The parameters used for simulation are the same as the milling conditions. The horizontal axis in the bifurcation diagram is the forcing frequency of the platform ranging from 16 to 32 Hz while the amplitude is fixed to 2.1 mm. The vertical axis represents the normalised impact phase of the ball. In Figure 2, the periodic releasing happens before the period doubling cascade occurs, where the ball starts to fly due to the inertial effect of the platform and, after a short period, collides with the platform due to the earth's gravitation and gets stuck after several low bounces. Since the vibratory ball milling system is highly damped (e = 0.15), the chaotic motion at the end of period doubling cascade is interrupted by the ball's landing in an absorbing region of the system, where the platform moves at the same direction as the ball. The second period doubling cascade may occur if the frequency increases continuously.
Figure 2: The bifurcation diagram of the bouncing
ball system generated under the milling conditions,
where e=0.15 and the amplitude of platform=2.1mm.
The strange attractor described in the time-delayed
Pseudo-Phase-Space (or the embedding phase space) was used to
identify the chaotic motion. The strange attractors discovered
at the end of the period doubling route to chaos for various
coefficients of restitution (e) are shown in Figure 3, where
and
are respectively signifying the time
intervals of two consecutive collisions.
Figure 3: Strange attractors obtained from simulation
for various coefficient of restitution, amplitude
= 0.1 mm,(a) e = 0.1; f = 92 .1 Hz, (b) e =0.3;
f = 78 Hz, (c) e = 0.6; f = 61.4 Hz, (d) e = 0.7;
f = 65.5 Hz, (e) e = 0.8; f = 60 Hz and (f) f = 0.86; f = 59.2 Hz.
It is seen that e determines the nature of the observed dynamics. Generally, the strange attractors can be classified as two types: highly damped or lightly damped. As shown in Figure 3(a), (b), the maps appear to collapse onto one or two curves. For such a highly damped system (e < 0.4), besides the periodic bouncing, chaos may occur while the ball collides with the platform always at the first or forth quadrants where the ball's bouncing can be maintained continuously. Otherwise, the ball will stick to the platform in the absorbing region, and the motion is then not chaotic, but rather quasiperiodic. Since the system is highly dissipative, the ball's trajectories are limited to a small range, which results in an attractor with a curve of highly organised points. For lightly damped systems (e > 0.7), the ball's bouncing can be attained at any phase of the platform due to very little energy loss during the collision and its trajectories may be very different. Therefore, the Pseudo-Phase-space maps of chaotic motions appear as a cloud of unorganised points in the phase plane (for example, Figure 3(e), (f)). For the systems between the highly and lightly damped ones (0.4< e < 0.7), the maps are between the above two attractors and appear as an infinite set of highly organised points in what appear to be a fractal-like structure (for example, Figure 3(d)), which is the typical feature of chaotic dynamics.
In terms of the correlation fractal dimension, the strangeness of the attractors can be estimated based on an improved Grassberger-Procaccia method [10]. The correlation dimensions are plotted as a function of e in Figure 4. The dimension generally increases with the decreasing e. But there is an apparent small difference in dimension between highly and lightly damped systems. Its value is about 1.6 for highly damped systems and about 1.8 for lightly damped ones. The results are in good agreement with the observed phenomena according to the physical meaning of the fractal dimension. It is noted that when e = 0.6, two attractors appear in the phase space (Figure 3(c)). Detailed analysis of the set of data shows that the lightly damped attractor first appeared and is semistable or transient, and then transformed into the highly damped attractor. Since the attractor which first appeared is incomplete and transient, the dimension should be estimated on the stable part of this system.
Figure 4: The correlation dimension is plotted as
a function of the coefficient of restitution.
The strange attractor which occurred in the vibratory ball milling system is shown in Figure 5(a). For comparison, the simulation result under the same conditions (where e = 0.15 obtained from free falling experiments) is given in Figure 5(b). The map obtained from simulation is collapsed onto one curve (Figure 5(b)), but the map obtained from the experiment (Figure 5(a)) is constructed by some random points which likely distributes along the simulated strange attractor. The external perturbations, such as the variation in forcing frequency and coefficient of restitution, are probably the reasons causing the noisy strange attractor. To support this explanation, the perturbations from the forcing frequency and e have been respectively introduced into the simulation by superposing a small sinusoidal variation to the forcing frequency and e. Their magnitudes are similar to those in the experimental ball milling system. The results from simulation are shown in Figure 5(c), (d). In Figure 5(c), the strange attractor with the perturbation of unstable forcing frequency becomes more random than that without perturbations and is similar to the experimental result. The perturbation from unstable e also results in a noisy attractor (Figure 5(d)), but its effect seems to be smaller than that from the frequency.
Figure 5: Strange attractors,
amplitude = 1.82 mm,
,
,
(a) measured from the experiment, (b) obtained from simulation
without any perturbation, (c) obtained from simulation with
perturbation of unstable frequency, that is
,
and (d) obtained from simulation with perturbation of unstable e,
that is 
.
Numerical analysis shows that the coefficient of restitution determines the observed dynamics of the bouncing ball system. Chaotic motion which occurred at the end of period doubling cascade for various e can be classified as two types in terms of strange attractors and their correlation dimensions are smaller than 2. Experimental measurements in a vibratory ball mill have qualitatively confirmed that chaotic motion occurred at the end of the period doubling route. However, the strange attractor of chaotic motion from the experiment appears to be somewhat different from that generated by the simulation due to external perturbations, such as the unstable platform velocity and coefficient of restitution.
The authors wish to thank A/Prof. B. Kenny and Mr. M. Young for useful suggestions and discussion. We also thank Dr. N. Tufillaro and Mr. M. Hamblin for permission to use their software for generating the bifurcation diagram and calculating the fractal dimensions.
An Investigation of Chaotic Phenomena in a Vibratory Ball Milling System
This document was generated using the LaTeX2HTML translator Version 96.1 (Feb 5, 1996) Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.
The command line arguments were:
latex2html han.
The translation was initiated by Pam Milliken on Wed Nov 27 16:06:12 EST 1996
