Hyperchaotic qualities of the ball motion in a ball milling device

Abstract

Ball collisions in milling devices are governed by complex dynamics ruled by impredictable impulsive forces. In this paper, nonlinear dynamics techniques are employed to analyze the time series describing the trajectory of a milling ball in an empty container obtained from a numerical model. The attractor underlying the system dynamics was reconstructed by the time delay method. In order to characterize the system dynamics the calculation of the spectrum of Lyapunov exponents was performed. Six Lyapunov exponents, divided into two terns with opposite sign, were obtained. The detection of the positive tern demonstrates the occurrence of the hyperchaotic qualities of the ball motion. A fractal Lyapunov dimension, equal to 5.62, was also obtained confirming the strange features of the attractor. © 1999 American Institute of Physics. ͓S1054-1500͑99͒00101-9͔The synthesis of metastable phases and new materials often requires severe processing conditions. Techniques based on heavy mechanical deformation of solid mixtures have reached a stage of considerable development in the field of materials science. Due to its commercial possibilities and to the unusual physical phenomena involved, powder processing by ball milling still arouses both technological and scientific interest. Energy is transferred from the milling tools to the entrapped powder, which is subject to heavy mechanical loads and undergoes continuous defect accumulation and an ultimate limit of stability is achieved. Diffusion and mass transport phenomena occur through the continuously developed fresh surfaces. The phase transformation and its rate are intimately linked to the energy transfer of the collision event. The efficiency of the transfer depends, in turn, on the degree of elasticity of the collision, which also affects the general features of the milling dynamics. As the elasticity increases, the milling dynamics change from periodic to aperiodic. This behavior can be suitably characterized with the help of nonlinear dynamical analyses. In this paper, a milling system was modeled and the trajectory of a single ball colliding elastically inside the milling container was numerically simulated. Nonlinear dynamics analyses showed that elastic impact conditions determine the occurrence of a hyper-chaotic behavior.

Onset of chaotic dynamics in a ball mill: Attractors merging and crisis induced intermittency

“…This attractor is responsible for a hyper-chaotic trajectory as already discussed in a previous paper. 7 As evident from Fig. 11, the system dynamics is so sensitive that two orbits originating from very similar initial conditions become distinguishable after less than 0.15 s.…”

“…Suitable nonlinear dynamics analysis proved that the ball undergoes an hyperchaotic dynamics characterized by three positive exponents in the Lyapunov spectrum. 7 In the present contribution, the analysis is pursued further and extended to all the possible impact conditions. The degree of impact elasticity is gradually changed and used as bifurcation parameter.…”

“…Preliminary investigations have already shown the possible occurrence of chaotic regimes during milling treatments. 6,7 On the other hand, modeling results and experimental evidences demonstrate the occurrence of regular dynamical regimes allowing for the direct measurement of both the average collision frequency and impact energy. 8 It becomes therefore important to study the transition from periodic to chaotic regimes in order to understand when and why transition takes place, so as to avoid it.…”

Milling criteria for the synthesis of nanocrystalline NiAl by mechanical alloying

“…Murty). a significant impetus to the study of milling dynamics [9][10][11][12][13], computer simulation of ball motion during ball milling [14], milling models [15,16] and milling maps [17,18]. The primary objective of these studies concerns the development of systematic prediction capability for the milling products during ball milling.…”

Multifractal properties of ball milling dynamics

“…2,6,7 For instance, the elasticity of collisions governs the transition from chaotic to periodic behaviors, thus allowing for the promotion of ordered patterns in the ball trajectories. 19,20 In this respect, it is worth noting that spatio-temporal organization in chaotic systems can also be sustained with a noise-driven stabilization of unstable orbits within the related attractors in the phase-space. 21 More in general, strange attractors that typically underlie chaotic dynamics embed themselves ordered and highly non-trivial patterns within their compact topology.…”