Fractal simulation of surface topography and prediction of its lubrication characteristics

A machined surface has observable fractal characteristics, with infinite local and overall self-similar consistency. Therefore, the fractal theory is considered to provide a better description of the morphological characteristics of rough surfaces, which accurately reflects the randomness and multi-scale characteristics of rough surfaces and it is not comparable with the surface characteristics obtained based on statistical parameters limited by sampling length and device resolution. In this study, the Weierstrass-Mandelbrot (W-M) function was applied to construct a fractal reconstruction surface, and the mixed elastohydrodynamic lubrication model was used to investigate the lubrication characteristics of real and reconstructed surfaces under the same fractal parameters. The effects of the fractal parameters on the fractal surface lubrication characteristics were further analyzed. The results demonstrate that the lateral roughness fractal surface provides greater resistance to the entrained flow of lubricant, which leads to a larger average film thickness, than the longitudinal roughness and isotropic fractal surface. With the increase in fractal dimension, the surface roughness peak density increases, which reduces the surface film thickness by 47%, and the friction coefficient increases by 46%. The lubrication parameter fluctuates slightly with the change in the number of overlapping ridges M of the fractal surface. Generally, M has little effect on the surface lubrication characteristics.

Fractal Characterization and Petrophysical Analysis of 3D Dynamic Digital Rocks of Sandstone

It is a crucial issue to comprehensively study the relations between microstructure and seepage capacity of porous media. Several physical-based parameters of fractal geometry can analyze the pore structure of rocks, while permeability and electrical conductivity are used to study seepage capacity. In this paper, we first created 3D dynamic digital models of nine different sandstones with varying clay content, cements, and intragranular pores in feldspar. These nine models were divided into three groups. Then, fractal dimension, lacunarity, and succolarity, permeability, and electrical properties of the models were calculated, and their relationships were investigated. We used fractal parameters to interpret the correlation between fluid flow and pore structure as one of the main petrophysical properties of a rock. Results showed that the coefficient of determination for cementation exponent m and fractal dimension is 0.869, while between m and porosity, and succolarity, it is 0.784 and 0.781, respectively. This indicates that the fractal dimension and cementation exponent describe the complexity of pores. The coefficient of determination between permeability and succolarity is 0.975, which is higher than that between permeability and the fractal dimension or porosity. The coefficient of determination between formation factor and succolarity is 0.957, which is higher than that between formation factor and the fractal dimension or porosity. Overall, a stronger relationship between petrophysical parameters, permeability in particular, and succolarity allows this lesser-used fractal parameter to be a good measure for characterizing the connectivity of pore space and pore network.

The Study of Usefulness of a Set of Fractal Parameters to Build Classes of Disease Units Based on Images of Pigmented Skin Lesions

Dermatoscopic images are also increasingly used to train artificial neural networks for the future to provide fully automatic diagnostic systems capable of determining the type of pigmented skin lesion. Therefore, fractal analysis was used in this study to measure the irregularity of pigmented skin lesion surfaces. This paper presents selected results from individual stages of preliminary processing of the dermatoscopic image on pigmented skin lesion, in which fractal analysis was used and referred to the effectiveness of classification by fuzzy or statistical methods. Classification of the first unsupervised stage was performed using the method of analysis of scatter graphs and the fuzzy method using the Kohonen network. The results of the Kohonen network learning process with an input vector consisting of eight elements prove that neuronal activation requires a larger learning set with greater differentiation. For the same training conditions, the final results are at a higher level and can be classified as weaker. Statistics of factor analysis were proposed, allowing for the reduction in variables, and the directions of further studies were indicated.

Development of Novel Fractal Method for Characterizing the Distribution of Blood Flow in Multi-Scale Vascular Tree

Blood perfusion is an important index for the function of the cardiovascular system and it can be indicated by the blood flow distribution in the vascular tree. As the blood flow in a vascular tree varies in a large range of scales and fractal analysis owns the ability to describe multi-scale properties, it is reasonable to apply fractal analysis to depict the blood flow distribution. The objective of this study is to establish fractal methods for analyzing the blood flow distribution which can be applied to real vascular trees. For this purpose, the modified methods in fractal geometry were applied and a special strategy was raised to make sure that these methods are applicable to an arbitrary vascular tree. The validation of the proposed methods on real arterial trees verified the ability of the produced parameters (fractal dimension and multifractal spectrum) in distinguishing the blood flow distribution under different physiological states. Furthermore, the physiological significance of the fractal parameters was investigated in two situations. For the first situation, the vascular tree was set as a perfect binary tree and the blood flow distribution was adjusted by the split ratio. As the split ratio of the vascular tree decreases, the fractal dimension decreases and the multifractal spectrum expands. The results indicate that both fractal parameters can quantify the degree of blood flow heterogeneity. While for the second situation, artificial vascular trees with different structures were constructed and the hemodynamics in these vascular trees was simulated. The results suggest that both the vascular structure and the blood flow distribution affect the fractal parameters for blood flow. The fractal dimension declares the integrated information about the heterogeneity of vascular structure and blood flow distribution. In contrast, the multifractal spectrum identifies the heterogeneity features in blood flow distribution or vascular structure by its width and height. The results verified that the proposed methods are capable of depicting the multi-scale features of the blood flow distribution in the vascular tree and further are potential for investigating vascular physiology.

Impact of vertical succussion strokes vs. vortex potentization on droplet evaporation patterns obtained from Iscador Quercus 3x potency

Background Pharmaceutical processing of homeopathic potencies consists of consecutively performed dilution and succussion steps. While the dilution steps are well defined, the manner of performing the succussions varies broadly among potency producers. Aims To study the impact of potentization consisting in the performance of vertical succussion strokes vs. vortex-like flow on droplet evaporation patterns obtained from Iscador Quercus 3x (ISCQ 3x). Methodology ISCQ 3x was prepared in three following variants: potentized for 2.5 min (i) by application of mechanically performed vertical strokes, or (ii) hand-made vortex-like flows; or (iii) only diluted and not-succussed control. Droplet evaporation method was performed as described in (1); in short: droplets of the three ISCQ 3x variants were evaporated on microscope slides (56 droplets of each variant distributed on four slides were evaporated in one experimental repetition). The experimental setup robustness was monitored by means of positive systematic control experiments, where on all 12 slides droplets of the ISCQ 3x variant potentized by the application of strokes were evaporated. The experiments were repeated five times. The resulting droplet residues were photographed in magnification 100x; the patterns were analyzed by means of the Image J software for their grey level distribution and textural and fractal parameters. Results and discussion All three ISCQ 3x variants could be significantly differentiated regarding some textural and fractal parameters; most parameters differentiated between the variant potentized by means of vertical strokes and the control and vortex-potentized variants. Fractal and textural parameters ranked the samples differently. Control experiments showed a reasonable experimental setup robustness. Conclusion The potentization by performing mechanical strokes vs. hand-made vortex-like flows influenced some phenomenological aspects of droplet evaporation patterns. This might indicate that some changes occurred on substance level as consequence of the mechanical impact. Further studies are necessary in this field.

Study on Gear Contact Stiffness and Backlash of Harmonic Drive Based on Fractal Theory

Contact stiffness and backlash model of harmonic reducer is related to robot’s positioning accuracy and vibration characteristics. Harmonic reducer tooth pair height is typically less than 1 mm. Thus, backlash and contact stiffness measurement and modeling are relatively complex. In this paper, contact stiffness and backlash model is proposed by establishing a relationship between fractal parameters and tooth contact load. Non-contact optical profiler and RMS method are combined to obtain fractal roughness parameters of real machined tooth surface. Finally, the effect of rough tooth surface and contact force fractal parameters on contact stiffness and gear backlash is studied. The results indicate that surface topography parameters and contact force have significant effects on contact stiffness and backlash. By increasing the fractal dimension, a decrease of gear backlash and contact stiffness is observed. However, the opposite is true for the fractal roughness parameter. Lastly, an increase in contact force improves the contact stiffness.

A Three-Dimensional Fractal Contact Model for a Rough Friction Surface With Multiple-Scale Asperities

This paper describes a new fractal contact model for a rough three-dimensional friction surface considering scale dependence. The model predicts the total contact area as a function of contact load. The microcontact fractal model for asperities at multiple-scale levels is derived. The truncated area distribution function of asperities at multiple-scale levels is revised. The effects of each scale level and fractal parameters on the proposed model are evaluated by numerical simulations. Results obtained from the complex fractal model with several scales are accurate as demonstrated by comparing them to experimental results and models taken from literature. Friction and wear of rough surfaces can be further studied using the proposed model.

Investigation of Stereometric and Fractal Patterns of Spin-Coated LuMnO3 Thin Films

In this paper, we have performed qualitative and quantitative analysis of LuMnO3 thin films surfaces, deposited by spin coating over Pt(111)/TiO2/SiO2/Si substrates, to evaluate their spatial patterns as a function of the film’s sintering temperature. Atomic force microscopy was employed to obtain topographic maps that were extensively analyzed via image processing techniques and mathematical tools. 3D (three-dimensional) topographical images revealed that films sintered at 650°C and 750°C presented the formation of smoother surfaces, while the film sintered at 850°C displayed a rougher surface with a root mean square roughness of ∼2.5 nm. On the other direction, the height distribution of the surface for all films has similar asymmetries and shape, although the film sintered using the highest temperature showed the lower density of rough peaks and a sharper peak shape. The advanced fractal parameters revealed that the film sintered at 850°C is dominated by low spatial frequencies, showing less spatial complexity, higher microtexture homogeneity, and uniform height distribution. These results suggest that the combination of stereometric and fractal parameters can be especially useful for identification of unique topographic spatial patterns in LuMnO3 thin films, helping in their implementation in technological applications, such as photovoltaic solar cells and information magnetic date storage and spintronic devices.