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The purpose of this study is to originally present noise analysis of electrical circuits defined on fractal set.

The fractal integrodifferential equations of resistor-inductor, resistor-capacitor, inductor-capacitor and resistor-inductor-capacitor circuits subjected to zero mean additive white Gaussian noise defined on fractal set have been formulated. The fractal time component has also been considered. The closed form expressions for crucial stochastic parameters of circuit responses have been derived from these equations. Numerical simulations of power spectral densities based on the derived autocorrelation functions have been performed. A comparison with those without fractal time component has been made.

We have found that the Hausdorff dimension of the middle b Cantor set strongly affects the power spectral densities; thus, the average powers of noise induced circuit responses and the inclusion of fractal time component causes significantly different analysis results besides the physical measurability of electrical quantities.

For the first time, the noise analysis of electrical circuit on fractal set has been performed. This is also the very first time that the fractal time component has been included in the fractal calculus-based circuit analysis.

This paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely useful for investigating the hidden behavior of the systems. The Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) derivatives are considered for the fractional structure of the model. Further, to add more complexity, the authors have taken the system with a CF fractal-fractional derivative having an exponential kernel. The active control technique is also considered for chaos control.

The systems under consideration are solved numerically. The authors show the Adams-type predictor-corrector scheme for the AB model and the Adams–Bashforth scheme for the CF model. The convergence and stability results are given for the numerical scheme. A numerical scheme for the FF model is also presented. Further, an active control scheme is used for chaos control and synchronization of the systems.

Simulations of the obtained solutions are displayed via graphics. The proposed system exhibits a very complex phenomenon known as chaos. The importance of the fractional and fractal order can be seen in the presented graphics. Furthermore, chaos control and synchronization between two identical fractional-order systems are achieved.

This paper mentioned the complex behavior of a dynamical system with fractional and fractal-fractional operators. Chaos control and synchronization using active control are also described.

Presents work carried out in the wider context of the IST Adrenalin project whose aim is to facilitate formation and lifecycle management of networked enterprises utilising concepts from two key information research areas. The approach described places heavy emphasis on the notion of mobile agent technologies and their adaptation for achieving the IT realisation of the above theoretical background. It covers specification, design and conceptual realisation of how information supply chains and routes can be organised and navigated across networked enterprise activities within the context of a branch independent model. Builds on the distribute and integrate concept as well as on the fractal idea by supporting self‐similarity, self‐organisation, self‐optimisation and dynamic organisational behavior. The harmonised combination of the concepts formulated in the Adrenalin theoretical framework and their IT realisation are employed within the context of a real‐world case study in an industrial ERP system.

An abstract lattice of empty set cells is shown to be able to account for a primary substrate in a physical space. Space‐time is represented by ordered sequences of topologically closed Poincaré sections of this primary space. These mappings are constrained to provide homeomorphic structures serving as frames of reference in order to account for the successive positions of any objects present in the system. Mappings from one section to the next involve morphisms of the general structures, representing a continuous reference frame, and morphisms of objects present in the various parts of this structure. The combination of these morphisms provides space‐time with the features of a non‐linear generalized convolution. Discrete properties of the lattice allow the prediction of scales at which microscopic to cosmic structures should occur. Deformations of primary cells by exchange of empty set cells allow a cell to be mapped into an image cell in the next section as far as the mapped cells remain homeomorphic. However, if a deformation involves a fractal transformation to objects, there occurs a change in the dimension of the cell and the homeomorphism is not conserved. Then, the fractal kernel stands for a “particle” and the reduction of its volume (together with an increase in its area up to infinity) is compensated by morphic changes of a finite number of surrounding cells. Quanta of distances and quanta of fractality are demonstrated. The interactions of a moving particle‐like deformation with the surrounding lattice involves a fractal decomposition process, which supports the existence and properties of previously postulated inerton clouds as associated to particles. Experimental evidence of the existence of inertons is reviewed and further possibilities of experimental proofs proposed.

This research investigated the mechanism of wet friction plates of engagement and solved the problem that the lock-up friction coefficient of sinter material could not be obtained but from experiments for a long time. The paper aims to discuss these issues.

Including four steps: surface topology sampling and reconstruction, fractal parameters obtaining and fractal surface simulating, micro-contact mechanics model and friction coefficient fractal model, and experimental verification.

After running in stage of the friction plates, the fractal dimension would reach a dynamically stable stage for a long time. The proportional coefficient K expresses the correlation between the base hardness and the asperities shear strength. The model could be property for one or more working condition via adjusting the coefficient K. The experiment data of friction coefficient are increased as the load magnified both in the model prediction and experiment practice. The trend is different from other models.

This research is original and it is supported by national defense project. It would be served for tracked vehicles to solve the defect in transmission system. The friction coefficient is obtained via solving the tangential force in MB model. The surface topography could be reconstructed by laser topography instrument and the parameters could be received by program.

The purpose of this paper is to investigate the influence of the resolution with which interfaces of fractal geometry are represented, on the contact area and consequently on the contact interfacial stresses. The study is based on a numerical approach. The paper focuses on the differences between the cases of elastic and inelastic materials having as primary parameter the resolution of the interface.

A multi‐resolution parametric analysis is performed for fractal interfaces dividing a plane structure into two parts. On these interfaces, unilateral contact conditions are assumed to hold. The computer‐generated surfaces adopted here are self‐affine curves, characterized by a precise value of the resolution δ of the fractal set. Different contact simulations are studied by applying a horizontal displacement s on the upper part of the structure. For every value of s, a solution is taken in terms of normal forces and displacements at the interface. The procedure is repeated for different values of the resolution δ. At each scale, a classical Euclidean problem is solved by using finite element models. In the limit of the finest resolution, fractal behaviour is achieved.

The paper leads to a number of interesting conclusions. In the case of linear elastic analysis, the contact area and, consequently, the contact interfacial stresses depend strongly on the resolution of the fractal interface. Contrary, in the case of inelastic analysis, this dependence is verified only for the lower resolution values. As the resolution becomes higher, the contact area tends to become independent from the resolution.

The originality of the paper lies on the results and the corresponding conclusions obtained for the case of inelastic material behaviour, while the results for the case of elastic analysis verify the findings of other researchers.

The purpose of this paper is to develop a general mathematic approach to model the microstructures of porous structures produced by additive manufacturing (AM), which will result in fractal surface topography and higher roughness that have greater influence on the performance of porous structures.

The overall shapes of pores were modeled by triply periodic minimal surface (TPMS), and the micro-roughness details attached to the overall pore shapes were represented by Weierstrass–Mandelbrot (W-M) fractal representation, which was integrated with TPMS along its normal vectors. An index roughly reflecting the irregularity of fractal TPMS was proposed, based on which the influence of the fractal parameters on the fractal TPMS was qualitatively analyzed. Two complex samples of real porous structures were given to demonstrate the feasibility of the model.

The fractal surface topography should not be neglected at a micro-scale level. In addition, a decrease in the fractal dimension D_s may exponentially make the topography rougher; an increase in the height-scaling parameter G may linearly increase the roughness; and the number of the superposed ridges has no distinct influence on the topography. Furthermore, the synthesis method is general for all implicit surfaces.

The method provides an alternative way to shift the posteriori design paradigm of porous media to priori design mode through numeric simulation. Therefore, the optimization of AM process parameters, as well as the porous structure, can be potentially realized according to specific functional requirement.

The synthesis of TPMS and W-M fractal geometry was accomplished efficiently and was general for all implicit freeform surfaces, and the influence of the fractal parameters on the fractal TPMS was analyzed more systematically.

The purpose of this paper is to reveal the dynamic contact characteristics of the slip ring. Dynamic contact resistance models considering wear and self-excited were established based on fractal theory.

The effects of tangential velocity, stiffness and damping coefficient on dynamic contact resistance are studied. The relationships between fractal parameters, wear time and contact parameters are revealed.

The results show that the total contact area decreases with the friction coefficient and fractal roughness under the same load. Self-excited vibration occurs at a low speed (less than 0.6 m/s). It transforms from stick-slip motion at 0.4 m/s to pure sliding at 0.5 m/s. A high stiffness makes contact resistance fluctuate violently, while increasing the damping coefficient can suppress the self-excited vibration and reduce the dynamic contact resistance. The fractal contact resistance model considering wear is established based on the fractal parameters models. The validity of the model is verified by the wear tests.

The results have a great significance to study the electrical contact behavior of conductive slip ring.

The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-09-2023-0300/

The distribution of the deformations of elementary cells is studied in an abstract lattice constructed from the existence of the empty set. One combination rule determining oriented sequences with continuity of set‐distance function in such spaces provides a particular kind of space‐time‐like structure which favors the aggregation of such deformations into fractal forms standing for massive objects. A correlative dilatation of space appears outside the aggregates. At large scale, this dilatation results in an apparent expansion, while at submicroscopic scale the families of fractal deformations give rise to families of particle‐like structure. The theory predicts the existence of classes of spin, charges and magnetic properties, while quantum properties associated with mass have previously been shown to determine the inert mass and the gravitational effects. When applied to our observable space‐time, the model would provide the justifications for the existence of the creation of mass in a specified kind of void, and the fractal properties of the embedding lattice extend the phenomenon to formal justifications of big‐bang‐like events without any need for supply of an extemporaneous energy.