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

In mechanical treatment carried out by ball milling, powder particles are subjected to repeated high-energy mechanical loads which induce heavy plastic deformations together with fracturing and cold-welding events. Owing to the continuous defect accumulation and interface renewal, both structural and chemical transformations occur. The nature and the rate of such transformations have been shown to depend on variables, such as impact velocity and collision frequency that depend, in turn, on the whole dynamics of the system. The characterization of the ball dynamics under different impact conditions is then to be considered a necessary step in order to gain a satisfactory control of the experimental set up. In this paper we investigate the motion of a ball in a milling device. Since the ball motion is governed by impulsive forces acting during each collision, no analytical expression for the complete ball trajectory can be obtained. In addition, mechanical systems exhibiting impacts are strongly nonlinear due to sudden changes of velocities at the instant of impact. Many different types of periodic and chaotic impact motions exist indeed even for simple systems with external periodic excitation forces. We present results of the analysis on the ball trajectory, obtained from a suitable numerical model, under growing degree of impact elasticity. A route to high dimensional chaos is obtained. Crisis and attractors merging are also found. (c) 2002 American Institute of Physics.     

Subharmonic bifurcations and chaotic dynamics of an air damping completely inelastic bouncing ball

We investigate the dynamics of a plastic ball on a vibrated platform in air by introducing air damping effect into the completely inelastic bouncing ball model. The air damping gives rise to larger saddle-node bifurcation points and a chaos confirmed by the largest Lyapunov exponent of a one-dimensional discrete mapping. The calculated bifurcation point distribution shows that the periodic motion of the ball is suppressed and a chaos emerges earlier for an increasing air damping. When the reset mechanism and the linear stability which cause periodic motion of the ball both collapse, the investigated system is fully chaotic.     

Preparation of nanocrystalline metal hydrides by ball milling

The diffusion of hydrogen into metals during ball milling has been investigated. A system is described that enables the milling to be done under approximately constant hydrogen pressure and allows for the continuous measurement of the quantity of hydrogen absorbed. This method has been used to prepare nanocrystalline hydrides of a variety of early transition metals and alloys. X-ray diffraction studies are used to identify phases present before and after milling and the width of diffraction peaks is used to estimate average grain size. Changes in the microstructure of Fe-containing samples during hydrogen absorption have been investigated by Mössbauer effect. Studies of the influence of milling conditions and the effect of milling time are also presented.     

Dynamics of a new composite four–Scroll chaotic system

A new three-dimensional autonomous chaotic system is proposed. This new system can generate single-scroll, double-scroll, three-scroll and four-scroll attractors under different system parameters. Particularly, it can generate a four-scroll chaotic attractor composed of a large Chua-like attractor and a small Lorenz-like attractor. And the system can also generate a nested three-scroll attractor and the multi-double-scroll chaotic attractor. In addition, the system possesses the chaotic state transition, and the number of scrolls will change in the state transition process. The formation mechanism of the composite four-scroll chaotic attractor is analyzed in detail. The dynamic analysis methods include time series, 0–1 test chart, phase diagram, bifurcation diagram and Lyapunov exponents are used to describe some basic dynamics behaviors of the proposed system.     

Iron nanoparticles produced by high-energy ball milling

In this investigation, the chemical and structural characteristics of Fe nanoparticles synthesized by high-energy ball milling have been explored. After the milling process the nanoparticles were collected using a magnetic field. The structure, morphology and composition of the powders were obtained using high-resolution electron microscopy. HREM images confirmed the nanoparticles’ presence with approximately 2–4 nm in size. It was found that using this method allowed the formation of nanoparticles in a smaller size range than other synthesis methods. Also, it was confirmed by HREM images that the obtained nanoparticles were mainly of the fcc nature and some of them of the MTP type.     

Study on the bonding state for carbon-boron nitrogen with different ball milling time

The varied bonding state and microstructure characterization were discussed for carbon-boron nitrogen (CBN) with abundant phase structure and nanostructure, which were synthesized directly by mechanical alloying technique at room temperature. According to the results of SEM and X-ray photoelectron spectroscopy (XPS) of CBN with different ball milling time, it is substantiated that the bonding state and microstructure for CBN were closely related to the ball milling time. With the increase of the ball milling time, some new chemical bonding states of CBN were observed, which implies that some new bonding state and microstructures have been formed. The results of XPS are accordance with that of X-ray diffraction of CBN.     

A novel 3D autonomous system with different multilayer chaotic attractors

This Letter proposes a novel three-dimensional autonomous system which has complex chaotic dynamics behaviors and gives analysis of novel system. More importantly, the novel system can generate three-layer chaotic attractor, four-layer chaotic attractor, five-layer chaotic attractor, multilayer chaotic attractor by choosing different parameters and initial condition. We analyze the new system by means of phase portraits, Lyapunov exponent spectrum, fractional dimension, bifurcation diagram and Poincaré maps of the system. The three-dimensional autonomous system is totally different from the well-known systems in previous work. The new multilayer chaotic attractors are also worth causing attention.     

A novel hyperchaotic system and its circuit implementation

A four-dimensional hyperchaotic system with five parameters is proposed. Its dynamical properties such as dissipativity, equilibrium points, Lyapunov exponent, Lyapunov dimension, bifurcation diagrams and Poincare maps are analyzed theoretically and numerically. Theoretical analyses and simulation tests indicate that the new system's dynamics behavior can be periodic attractor, chaotic attractor and hyperchaotic attractor as the parameter varies. Finally, the circuit of this new hyperchaotic system is designed and realized by Multisim software. The simulation results confirm that the chaotic system is different from the existing chaotic systems and is a novel hyperchaotic system. The system is recommendable for many engineering applications such as information processing, cryptology, secure communications, etc.     

Simulation of ball motion and energy transfer in a planetary ball mill

A kinetic model is proposed for simulating the trajectory of a single milling ball in a planetary ball mill, and a model is also proposed for simulating the local energy transfer during the ball milling process under no-slip conditions. Based on the kinematics of ball motion, the collision frequency and power are described, and the normal impact forces and effective power are derived from analyses of collision geometry. The Hertzian impact theory is applied to formulate these models after having established some relationships among the geometric, dynamic, and thermophysical parameters. Simulation is carried out based on two models, and the effects of the rotation velocity of the planetary disk and the vial-to-disk speed ratio ω/Ω on other kinetic parameters is investigated. As a result, the optimal ratio ω/Ω to obtain high impact energy in the standard operating condition at = 800 rpm is estimated, and is equal to 1.15.     

A new hyperchaos system and its circuit simulation by EWB

This paper reports a new hyperchaotic system evolved from the three-dimensional Lü chaotic system. The Lyapunov exponents spectrum and the bifurcation diagram of this new hyperchaotic system are obtained. Hyperchaotic attractor, periodic orbit and chaotic attractor are obtained by computer simulation. A circuit is designed to realize this new hyperchaotic system by electronic workbench.     

The influence of ball milling on the structure and electroluminescence of zinc sulphide powder

The paper describes the influence of ball milling on the structure and electroluminescent emittance of zinc sulphide powder activated with manganese and copper. It is shown that in the initial stage of ball milling there is a sudden decrease in the EL emittance and later the EL emittance fluctuates. The maximum in fluctuation, however, does not exceed 25% of the initial emittance. The ball milling mechanism of powder and the quenching of its EL emittance was proposed on the basis of measurements obtained by X-ray, spectral and integrated EL emittance as a function of the milling time. From Goldberg's relation for EL emittance a new relation was obtained for the emittance as a function of the structural parameters such as the magnitude of the powder grain, its so-called critical value, and the density of imperfections produced during milling.The author thanks J. Waková from the Research Institute of Inorganic Chemistry in Ústí nad Labem for making it possible to perform the diffractometric measurements in their laboratory, J. Pasterák from the Institute of Physics, Czechoslovak Academy of Sciences, who made it possible to measure the EL spectral emittance, Prof. A. Kochanovská for advice in evaluating the X-ray results, M. Trlifaj, K. Pátek and J. Kubátová for advice in interpreting the EL spectral emittance and A. Kleprlík for help in elaborating the results.     

Correlation between jerky flow and jerky dynamics in a nanoscratch on a metallic glass film

Chaos has been well understood in dynamic system, however, how the chaotic behavior occur in jerky flow in material, is still not clear, and is lack of specific chaotic attractor. Here the jerky evolution of lateral force and the stair-like fluctuation of lateral displacement are observed for Ni62 Nb38(at.%) metallic glass film during nanoscratch process. This jerky flow is investigated by using the largest Lyapunov exponent, Kolmogorov entropy and fractal dimension, and chaotic behavior of lateral force-time and normal displacement-lateral displacement sequences is verified. In addition to time series analysis, it is found that jerk equation can be used to describe the jerky flow of the metallic-glass film during nanoscratch. More importantly, unambiguous chaotic attractor is presented by jerky dynamics using "jerk"-singularities, namely the total change rate of lateral force relative to scratch time. These reveal an inner connection between jerky flow and jerky dynamics in nanoscratch of a metallic-glass film.     

A novel four-wing non-equilibrium chaotic system and its circuit implementation

In this paper, we construct a novel, 4D smooth autonomous system. Compared to the existing chaotic systems, the most attractive point is that this system does not display any equilibria, but can still exhibit four-wing chaotic attractors. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, bifurcation diagram, and Poincaré maps. There is little difference between this chaotic system without equilibria and other chaotic systems with equilibria shown by phase portraits and Lyapunov exponents. But the bifurcation diagram shows that the chaotic systems without equilibria do not have characteristics such as pitchfork bifurcation, Hopf bifurcation etc. which are common to the normal chaotic systems. The Poincaré maps show that this system is a four-wing chaotic system with more complicated dynamics. Moreover, the physical existence of the four-wing chaotic attractor without equilibria is verified by an electronic circuit.     

一个具有完全四翼形式的四维混沌

本文通过引进一个非线性状态反馈控制器, 提出了一个新的四维混沌系统, 该混沌吸引子能在任何方向上都表现出四翼形式. 由于存在一个大的正李雅普诺夫指数, 混沌系统具有一些非常有趣和复杂的动力学行为. 对系统的一些基本动力学特性进行了数值模拟和理论分析, 如平衡点、耗散性、Poincaré映射、频谱、时域谱和混沌行为等. 通过对Lyapunov指数谱和分岔图的分析, 进一步研究了混沌行为的系统参数敏感性. 最后, 设计了一个实现四翼混沌系统的振荡电路, EWB观察结果与数值模拟结果具有良好的一致性.     

Irregular diffusion in the bouncing ball billiard

We call a system bouncing ball billiard if it consists of a particle that is subject to a constant vertical force and bounces inelastically on a one-dimensional vibrating periodically corrugated floor. Here we choose circular scatterers that are very shallow, hence this billiard is a deterministic diffusive version of the well-known bouncing ball problem on a flat vibrating plate. Computer simulations show that the diffusion coefficient of this system is a highly irregular function of the vibration frequency exhibiting pronounced maxima whenever there are resonances between the vibration frequency and the average time of flight of a particle. In addition, there exist irregularities on finer scales that are due to higher-order dynamical correlations pointing towards a fractal structure of this curve. We analyze the diffusive dynamics by classifying the attracting sets and by working out a simple random walk approximation for diffusion, which is systematically refined by using a Green–Kubo formula.     

Mixing of iron with various metals by high-energy ball milling of elemental powder mixtures

The alloying of Fe with T=V, Cr and Mn by high-energy ball milling of elemental powder mixtures has been studied from the scale of a powder particle down to the atomic scale using X-ray and neutron diffraction, Mössbauer spectrometry and magnetic measurements for Fe_1?x T _x alloys with x=0.50, 0.65 for T=V, x=0.50, 0.70 for T=Cr and x=0.72 for T=Mn. Different alloying behaviours are observed according to T once powder particles have the final composition. The rather fast mechanical alloying of Fe with Mn reflects the statistical nature of the milling process in contrast to the slow mixing of Fe with V and of Fe with Cr. Hyperfine magnetic field distributions remain stationary in shape in the last milling stage at room temperature both for T=V and T=Cr. Magnetic measurements evidence the persistence with milling time of a large population of nanometer-sized Fe-Cr zones that are superparamagnetic at room temperature and at 400 K. By contrast, room-temperature Mössbauer spectra show only a single line for long milling times. The unmixed stationary state of milled p-Fe_0.7Cr_0.3 is discussed in the light of a recent model of systems driven by competing dynamics.     

Synthesis and characterization of magnetite-maghemite nanoparticles obtained by the high-energy ball milling method

We present the process of synthesis and characterization of magnetite-maghemite nanoparticles by the ball milling method. The particles were synthesized in a planetary ball mill equipped with vials and balls of tempered steel, employing dry and wet conditions. For dry milling, we employed microstructured analytical-grade hematite (α-Fe₂O₃), while for wet milling, we mixed hematite and deionized water. Milling products were characterized by X-ray diffraction, transmission electron microscopy, room temperature Mössbauer spectroscopy, vibrating sample magnetometry, and atomic absorption spectroscopy. The Mössbauer spectrum of the dry milling product was well fitted with two sextets of hematite, while the spectrum of the wet milling product was well fitted with three sextets of spinel phase. X-ray measurements confirmed the phases identified by Mössbauer spectroscopy in both milling conditions and a reduction in the crystallinity of the dry milling product. TEM measurements showed that the products of dry milling for 100 h and wet milling for 24 h consist of aggregates of nanoparticles distributed in size, with mean particle size of 10 and 15 nm, respectively. Magnetization measurements of the wet milling product showed little coercivity and a saturation magnetization around 69 emu g^?1, characteristic of a nano-spinel system. Atomic absorption measurements showed that the chromium contamination in the wet milling product is approximately two orders of magnitude greater than that found in the dry milling product for 24 h, indicating that the material of the milling bodies, liberated more widely in wet conditions, plays an important role in the conversion hematite-spinel phase.     

A special type of attractor and transitions in a delayed system

We discuss strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors in a quasiperiodically driven system with time delays. A route and the associated mechanism are described for a special type of attractor called strange-nonchaotic-attractor-like (SNA-like) through T² torus bifurcation. The type of attractor can be observed in large parameter domains and it is easily mistaken for a true SNA judging merely from the phase portrait, power spectrum and the largest Lyapunov exponent. SNA-like attractor is not strange and has no phase sensitivity. Conditions for Neimark-Sacker bifurcation are obtained by theoretical analysis for the unforced system. Complicated and interesting dynamical transitions are investigated among the different tongues.     

Athermal character of the solid state amorphization of lactose induced by ball milling

In this paper, we report the possibility to reach pure glassy amorphous lactose by ball milling of crystalline α lactose under a dry nitrogen atmosphere. This route to the glassy state is found to be free of mutarotation towards the anomer β while this mutarotation is unavoidable using the usual thermal route, i.e. the quench of the liquid. This result definitely makes the ‘local quench melting’ hypothesis unsuitable to account for amorphization by ball milling.     

Sensitivity to initial conditions and lyapunov exponent of a quasiperiodic system

The dynamics of a quasiperiodic map is analyzed both in the presence and in the absence of weak noise. It is shown that, in the presence of weak noise, a strange chaotic attractor with a negative Lyapunov exponent and sensitive dependence of trajectories on the initial conditions can exist in the system. This means that the types of motion of a fluctuating system cannot be classified only by the sign of the leading Lyapunov exponent.